Finite Group Databasebeta

← C67×#19406 #30698657

Subgroups — C67×#19406 #30698657

Order 17152.

598 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C67×#3228C211yesmaximal, characteristic
2C67×#3112C211yesmaximal, characteristic
3C67×#3104C211yesmaximal, characteristic, complemented
4C67×#2982C211yesmaximal, characteristic, complemented
5C67×#1682C212yesmaximal, complemented
6C67×#1678C212yesmaximal, complemented
7C67×#1676C212yesmaximal, complemented
8C67×#1672C212yesmaximal, complemented
9C67×#1572C211yesmaximal, characteristic, complemented
10C67×#1554C211yesmaximal, characteristic, complemented
11C67×#1543C211yesmaximal, characteristic
12#533×C67C2211yescharacteristic
13#532×C67C2211yescharacteristic
14C2×C4×Q8×C67C2211yescharacteristic
15#500×C67C2211yescharacteristic
16#498×C67C2211yescharacteristic
17#486×C67C2211yescharacteristic
18#486×C67C2211yescharacteristic
19#484×C67—21no—
20#482×C67—22no—
21#480×C67—21no—
22#478×C67—22no—
23#477×C67C2212yes—
24#477×C67C2212yes—
25#462×C67C2212yes—
26#460×C67C2212yes—
27#451×C67—21no—
28#448×C67—21no—
29C8×Q8×C67—22no—
30C8×Q8×C67C2212yes—
31C8×Q8×C67C2212yes—
32#440×C67—22no—
33C4×Q16×C67—22no—
34#432×C67C2211yescharacteristic
35#431×C67—22no—
36#431×C67C2211yescharacteristic
37#431×C67C2211yescharacteristic
38#431×C67C2211yescharacteristic
39#335×C67C2212yes—
40#334×C67C2212yes—
41#333×C67C2212yes—
42#333×C67C2212yes—
43#332×C67C2212yes—
44#331×C67C2212yes—
45C2×#41×C67—21no—
46C22×Q8×C67D411yescharacteristic
47C2×Q16×C67—22no—
48C2×SD16×C67—22no—
49C4⋊Q8×C67C2311yescharacteristic
50C4⋊Q8×C67C2311yescharacteristic
51C4⋊Q8×C67C2311yescharacteristic
52#127×C67C2311yescharacteristic
53#125×C67—21no—
54#124×C67—21no—
55C22⋊Q8×C67—21no—
56C22⋊Q8×C67—21no—
57#121×C67—21no—
58#121×C67—21no—
59C4×Q8×C67—21no—
60C4×Q8×C67—21no—
61C4×Q8×C67—21no—
62C4×Q8×C67—21no—
63C4×Q8×C67D412yes—
64C4×Q8×C67C2312yes—
65C4×D4×C67—21no—
66C4×D4×C67—21no—
67#117×C67—21no—
68C2×C4⋊C4×C67—21no—
69C2×C4⋊C4×C67D411yescharacteristic
70C2×C42×C67—21no—
71C2×C42×C67C2311yescharacteristic
72#107×C67—22no—
73#107×C67—22no—
74#107×C67—22no—
75#106×C67—22no—
76C4⋊C8×C67—22no—
77C4⋊C8×C67—22no—
78C4⋊C8×C67D412yes—
79C4⋊C8×C67C2312yes—
80C4⋊C8×C67C2312yes—
81C4⋊C8×C67C2312yes—
82#103×C67—22no—
83#103×C67—22no—
84#103×C67—22no—
85#102×C67—22no—
86#102×C67—22no—
87#102×C67—22no—
88C22⋊C8×C67—21no—
89C22⋊C8×C67—21no—
90C22⋊C8×C67—21no—
91C22⋊C8×C67D412yes—
92#97×C67—21no—
93C4×C8×C67—22no—
94C4×C8×C67—22no—
95C4×C8×C67—21no—
96C4×C8×C67—21no—
97C4×C8×C67C2311yescharacteristic
98C4×C8×C67C2311yescharacteristic
99#41×C67—82no—
100C2×Q8×C67—24no—
101C2×Q8×C67—21no—
102C2×Q8×C67—21no—
103C2×Q8×C67—21no—
104C2×Q8×C67—21no—
105C2×Q8×C67C2×D412yes—
106C2×D4×C67—21no—
107C2×D4×C67—21no—
108C2×D4×C67—21no—
109C2×D4×C67—41no—
110C22×C4×C67—21no—
111C22×C4×C67—21no—
112C22×C4×C67—41no—
113C22×C4×C67C2×D411yescharacteristic
114C22×C4×C67C2×D411yescharacteristic
115C22×C4×C67#4111yescharacteristic
116Q16×C67—44no—
117SD16×C67—44no—
118C2×C8×C67—22no—
119C2×C8×C67—22no—
120C2×C8×C67—22no—
121C2×C8×C67C2×D412yes—
122C2×C8×C67C2×D412yes—
123C4⋊C4×C67—22no—
124C4⋊C4×C67—21no—
125C4⋊C4×C67—22no—
126C4⋊C4×C67—21no—
127C4⋊C4×C67—21no—
128C4⋊C4×C67—21no—
129C4⋊C4×C67—21no—
130C4⋊C4×C67—21no—
131C4⋊C4×C67—21no—
132C4⋊C4×C67—41no—
133C4⋊C4×C67—41no—
134C4⋊C4×C67C2×D412yes—
135C22⋊C4×C67—41no—
136C22⋊C4×C67—41no—
137C22⋊C4×C67—41no—
138C42×C67—22no—
139C42×C67—22no—
140C42×C67—41no—
141C42×C67C2×D411yescharacteristic
142C42×C67C2×D411yescharacteristic
143C42×C67#4111yescharacteristic
144C42×C67C2411yescharacteristic
145C23×C67—41no—
146C23×C67#12111yescharacteristic
147Q8×C67—24no—
148Q8×C67—22no—
149Q8×C67—42no—
150Q8×C67—81no—
151Q8×C67—81no—
152Q8×C67#13514yes—
153D4×C67—42no—
154D4×C67—42no—
155D4×C67—81no—
156D4×C67—82no—
157C2×C4×C67—24no—
158C2×C4×C67—22no—
159C2×C4×C67—22no—
160C2×C4×C67—22no—
161C2×C4×C67—22no—
162C2×C4×C67—41no—
163C2×C4×C67—41no—
164C2×C4×C67—41no—
165C2×C4×C67—41no—
166C2×C4×C67—82no—
167C2×C4×C67—81no—
168C2×C4×C67#12111yescharacteristic
169C2×C4×C67#12111yescharacteristic
170C2×C4×C67#12111yescharacteristic
171C2×C4×C67C22×D411yescharacteristic
172C2×C4×C67C22×D411yescharacteristic
173C2×C4×C67C2×#4111yescharacteristic
174C8×C67—22no—
175C8×C67—22no—
176C8×C67—42no—
177C8×C67—42no—
178C8×C67—42no—
179C22×C67—21no—
180C22×C67—21no—
181C22×C67—21no—
182C22×C67—81no—
183C22×C67—81no—
184C22×C67—81no—
185C22×C67C2×#12111yescharacteristic
186C4×C67—21no—
187C4×C67—24no—
188C4×C67—21no—
189C4×C67—21no—
190C4×C67—21no—
191C4×C67—42no—
192C4×C67—81no—
193C4×C67—81no—
194C4×C67—81no—
195C4×C67C2×#13512yes—
196C4×C67#57812yes—
197#19406C6711yesmaximal, characteristic, direct product factor, Sylow
198C2×C67—41no—
199C2×C67—161no—
200C2×C67#304411yescharacteristic
201C2×C67#305011yescharacteristic
202C2×C67#306811yescharacteristic
203#3228C2×C6711yescharacteristic
204#3112C2×C6711yescharacteristic
205#3104C2×C6711yescharacteristic, complemented
206#2982C2×C6711yescharacteristic, complemented
207#1682C2×C6712yescomplemented
208#1678C2×C6712yescomplemented
209#1676C2×C6712yescomplemented
210#1672C2×C6712yescomplemented
211#1572C2×C6711yescharacteristic, complemented
212#1554C2×C6711yescharacteristic, complemented
213#1543C2×C6711yescharacteristic
214C67#1940611yescharacteristic, direct product factor, Sylow
215#533C22×C6711yescharacteristic
216#532C22×C6711yescharacteristic
217C2×C4×Q8C22×C6711yescharacteristic
218#500C22×C6711yescharacteristic
219#498C22×C6711yescharacteristic
220#486C22×C6711yescharacteristic
221#486C22×C6711yescharacteristic
222#484—21no—
223#482—22no—
224#480—21no—
225#478—22no—
226#477C22×C6712yes—
227#477C22×C6712yes—
228#462C22×C6712yes—
229#460C22×C6712yes—
230#451—21no—
231#448—21no—
232C8×Q8—22no—
233C8×Q8C22×C6712yes—
234C8×Q8C22×C6712yes—
235#440—22no—
236C4×Q16—22no—
237#432C22×C6711yescharacteristic
238#431—22no—
239#431C22×C6711yescharacteristic
240#431C22×C6711yescharacteristic
241#431C22×C6711yescharacteristic
242#335C22×C6712yes—
243#334C22×C6712yes—
244#333C22×C6712yes—
245#333C22×C6712yes—
246#332C22×C6712yes—
247#331C22×C6712yes—
248C2×#41—21no—
249C22×Q8D4×C6711yescharacteristic
250C2×Q16—22no—
251C2×SD16—22no—
252C4⋊Q8C23×C6711yescharacteristic
253C4⋊Q8C23×C6711yescharacteristic
254C4⋊Q8C23×C6711yescharacteristic
255#127C23×C6711yescharacteristic
256#125—21no—
257#124—21no—
258C22⋊Q8—21no—
259C22⋊Q8—21no—
260#121—21no—
261#121—21no—
262C4×Q8—21no—
263C4×Q8—21no—
264C4×Q8—21no—
265C4×Q8—21no—
266C4×Q8D4×C6712yes—
267C4×Q8C23×C6712yes—
268C4×D4—21no—
269C4×D4—21no—
270#117—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4D4×C6711yescharacteristic
273C2×C42—21no—
274C2×C42C23×C6711yescharacteristic
275#107—22no—
276#107—22no—
277#107—22no—
278#106—22no—
279C4⋊C8—22no—
280C4⋊C8—22no—
281C4⋊C8D4×C6712yes—
282C4⋊C8C23×C6712yes—
283C4⋊C8C23×C6712yes—
284C4⋊C8C23×C6712yes—
285#103—22no—
286#103—22no—
287#103—22no—
288#102—22no—
289#102—22no—
290#102—22no—
291C22⋊C8—21no—
292C22⋊C8—21no—
293C22⋊C8—21no—
294C22⋊C8D4×C6712yes—
295#97—21no—
296C4×C8—22no—
297C4×C8—22no—
298C4×C8—21no—
299C4×C8—21no—
300C4×C8C23×C6711yescharacteristic
301C4×C8C23×C6711yescharacteristic
302#41—82no—
303C2×Q8—24no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8C2×D4×C6712yes—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—41no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—41no—
316C22×C4C2×D4×C6711yescharacteristic
317C22×C4C2×D4×C6711yescharacteristic
318C22×C4#41×C6711yescharacteristic
319Q16—44no—
320SD16—44no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8C2×D4×C6712yes—
325C2×C8C2×D4×C6712yes—
326C4⋊C4—22no—
327C4⋊C4—21no—
328C4⋊C4—22no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4C2×D4×C6712yes—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C42—22no—
342C42—22no—
343C42—41no—
344C42C2×D4×C6711yescharacteristic
345C42C2×D4×C6711yescharacteristic
346C42#41×C6711yescharacteristic
347C42C24×C6711yescharacteristic
348C23—41no—
349C23#121×C6711yescharacteristic
350Q8—24no—
351Q8—22no—
352Q8—42no—
353Q8—81no—
354Q8—81no—
355Q8#135×C6714yes—
356D4—42no—
357D4—42no—
358D4—81no—
359D4—82no—
360C2×C4—24no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—82no—
370C2×C4—81no—
371C2×C4#121×C6711yescharacteristic
372C2×C4#121×C6711yescharacteristic
373C2×C4#121×C6711yescharacteristic
374C2×C4C22×D4×C6711yescharacteristic
375C2×C4C22×D4×C6711yescharacteristic
376C2×C4C2×#41×C6711yescharacteristic
377C8—22no—
378C8—22no—
379C8—42no—
380C8—42no—
381C8—42no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22C2×#121×C6711yescharacteristic
389C4—21no—
390C4—24no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—42no—
395C4—81no—
396C4—81no—
397C4—81no—
398C4C2×#135×C6712yes—
399C4#578×C6712yes—
400C2—41no—
401C2—161no—
402C2C67×#304411yescharacteristic
403C2C67×#305011yescharacteristic
404C2C67×#306811yescharacteristic
subgroupG/H
center, socleC22×C67C2×#121
commutator, Frattini, cosocleC42C24×C67
prime radical#532×C67C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C67 ◁ C2×C42×C67 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C67×#3104C2semidirect
4. C67×#2982C2semidirect
5. C67×#1682C2semidirect
6. C67×#1678C2semidirect
7. C67×#1676C2semidirect
8. C67×#1672C2semidirect
9. C67×#1572C2semidirect
10. C67×#1554C2semidirect
197. #19406C67direct (N × H)
205. #3104C2×C67semidirect
206. #2982C2×C67semidirect
207. #1682C2×C67semidirect
208. #1678C2×C67semidirect
209. #1676C2×C67semidirect
210. #1672C2×C67semidirect
211. #1572C2×C67semidirect
212. #1554C2×C67semidirect
214. C67#19406direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C67×#32282
C67×#31122
C67×#31042
C67×#29822
C67×#16822
C67×#16782
C67×#16762
C67×#16722
C67×#15722
C67×#15542
C67×#15432
#1940667